Direct Exchange Interactions in Solids
In a crystal, generalization of the Heisenberg Hamiltonian in which the sum is taken over the exchange Hamiltonians for all the (i,j) pairs of atoms of the many-electron system gives:.
-
(14)
The 1/2 factor is introduced because the interaction between the same two atoms is counted twice in performing the sums. Note that the J in Eq.(14) is the exchange constant Jab above not the exchange integral Jex. The exchange integral Jex is related to yet another quantity, called the exchange stiffness constant (A) which serves as a characteristic of a ferromagnetic material. The relationship is dependent on the crystal structure. For a simple cubic lattice with lattice parameter ,
-
(15)
For a body-centered cubic lattice,
-
(16)
and for a face-centered cubic lattice,
-
(17)
The form of Eq. (14) corresponds identically to the Ising model of ferromagnetism except that in the Ising model, the dot product of the two spin angular momenta is replaced by the scalar product SijSji. The Ising model was invented by Wilhelm Lenz in 1920 and solved for the one-dimensional case by his doctoral student Ernst Ising in 1925. The energy of the Ising model is defined to be:
-
(18)
Read more about this topic: Exchange Interaction
Famous quotes containing the words direct, exchange and/or interactions:
“Authority and power are two different things: power is the force by means of which you can oblige others to obey you. Authority is the right to direct and command, to be listened to or obeyed by others. Authority requests power. Power without authority is tyranny.”
—Jacques Maritain (1882–1973)
“To coöperate in the highest as well as the lowest sense, means to get our living together. I heard it proposed lately that two young men should travel together over the world, the one without money, earning his means as he went, before the mast and behind the plow, the other carrying a bill of exchange in his pocket. It was easy to see that they could not long be companions or coöperate, since one would not operate at all. They would part at the first interesting crisis in their adventures.”
—Henry David Thoreau (1817–1862)
“The exercise of power is determined by thousands of interactions between the world of the powerful and that of the powerless, all the more so because these worlds are never divided by a sharp line: everyone has a small part of himself in both.”
—Václav Havel (b. 1936)